Calculating damping for belts

ABSTRACT

A method of determining damping coefficients for a resilient web, involving: stretching a prescribed length of the web between a pair of pulleys mounted on a frame; attaching a prescribed test mass M o  upon this length at a prescribed distance L 1  from one pulley; shaking the frame at resonance frequency f R , and the while deriving resonance-amplitude; and using the foregoing to determine web damping coefficients.

This is a division of U.S. Ser. No. 08/418,221, filed Apr. 6, 1995 and issuing as U.S. Pat. No. 5,554,807 on Sep. 10, 1996.

BACKGROUND FEATURES

Workers making or using flexible web means (e.g., belts used to transmit motion from one shaft to another) have long been concerned about the complications now typically associated with determining damping characteristics (e.g., damping coefficient) of such a web. Typically, this may involve "trial-and-error" testing of the actual belt in the actual mechanism; or it may require a specific sample size, or it may require that the belt be permanently altered in some way; or "over-stressed" in actual use-environment.

This invention addresses such concerns, and teaches a technique and apparatus for determining such damping coefficients:

without need of trial-error testing of an actual belt length in an actual use-environment, or any associated belt altering or overstressing;

without need of any specific sample size or belt-length (e.g., testing one length can determine damping for many different lengths).

Thus, it is an object hereof to alleviate such problems and provide at least some of the here-described features and advantages. A more particular object is to provide means for quantifying belt damping parameters--especially for various belt-lengths, yet by testing at only a few belt-length positions. Another object is to do so by subjecting the belt to sinusoidal shaking at resonance conditions.

A further object is to avoid conventional solutions, such as testing a belt in the mechanism it is to be used in.

Other objects and advantages of the present invention will be apparent to those skilled in the art.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other features and advantages of the present invention will be appreciated by workers as they become better understood by reference to the following detailed description of the present preferred embodiments, these being considered in conjunction with the accompanying drawings, wherein like reference symbols denote like elements:

FIG. 1 is a very schematic, idealized showing of a preferred belt tensioning/shaking arrangement;

FIG. 2 illustrates a preferred measurement array for measuring belt damping for an arrangement like that of FIG. 1; and

FIGS. 3 and 4 illustrate two models for related analysis; and FIGS. 3A, 3B give related analysis.

DETAILED DESCRIPTION

FIG. 1 shows a frame F which is relatively rigid when compared to the belt B being tested, being secured to a shaker table (not shown) adapted to experience a known sinusoidal motion. Pulleys P-1, P-2 are affixed on the frame F. A length of belt B (endless-loop or not; also includes any belt-like, web material) is stretched over, and between, the pulleys P-1, P-2. The distance L between pulleys is adjustable so that a desired tension may be produced in belt length B. Such tensioning is well known to those practiced in the related arts. Belt B is clamped to the pulleys over a narrow arc, away from the points where the belt segments are tangent to the pulleys. This prevents gross relative motion between the belt and the pulleys.

A weight 10 of known mass (test mass m), is clamped to belt B at some known distance L1 from the center of one pulley P-2. Various known means of clamping may be used, such as a common C-clamp as known in the art. This clamp, with any needed transducer attached to it, should have its center of gravity near the (width, and thickness) axis of belt B.

Motion-Detect means MD (e.g., see accelerometer 2a, FIG. 2) is used for measuring the vertical motion-amplitude of mass m, either relative to the shaker table or relative to the same reference as the shaker motion. This may be accomplished by any one of many well known expedients such as: an accelerometer or like transducer (with indicator also used, if desired--see FIG. 2), or a recording pen, or a light beam reflected off the mass m onto a scale, or an electromagnetic voltage generator, etc. For illustration purposes, accelerometers will be preferred here to measure motions of both the mass m and the shaker table. Such accelerometers are well known to artisans familiar with motion transducers. For illustration purposes, these accelerometers are assumed to measure acceleration relative to the earth. Piezo-electric accelerometers are well known examples and preferred. Or, one may equivalently measure velocity or displacement, instead of acceleration, as artisans understand.

The belt and frame assembly is mounted to the shaker table so that the length of the belt L, under test, lies along the same direction as the motion of the shaker table (see arrow FIG. 1). Motion of the shaker table is transmitted to the mass m (along plane of frame F) via pulleys P-1, P-2 and belt B wound thereon.

The motion of mass m may, or may not, have the same amplitude as that of the shaker table because of the longitudinal (i.e.,; tension and compression) flexibility of the belt B. Belt B and object 10 clamped thereon may be viewed as, essentially, the equivalent of a spring and mass system, having a resonant frequency f_(R). We will here assume that when a spring and mass system is shaken at its resonant frequency, the motion of the mass is limited only by the damping in the system. Here, belt B is assumed to provide the only significant loss of energy in this system, thus damping must be, essentially, entirely due to the (dynamic) belt stretching. Here, we will assume that a damping coefficient c, in units of force per unit velocity, can be derived by the following equation: ##EQU1## Where: pi=3.14159

m: mass of the object clamped to the belt B

f_(R) : frequency of sinusoidal motion at resonance

xs: absolute value of motion amplitude for shaker table

xm: absolute value of amplitude for motion of mass m,

System for Measuring Damping (FIG. 2):

FIG. 2 is a block diagram of a preferred, related instrumentation system IS for determining belt damping-coefficient C. A tunable sine wave generator 2-1 is used to output a sine wave voltage at a selected frequency, (this may be monitored by associated frequency meter 2-3, and this sine wave voltage from generator 2-1 may be input to an amplifier 2-5). Generator 2-1 is coupled to drive an electrodynamic shaker table st. (This may alternatively be a electrohydraulically-driven shaker table.) The shaker table motion is preferably detected by an accelerometer 2-a (or like amplitude transducer means) whose electrical output may be fed (e.g., via an amplifier 2-7 if needed), to an amplitude meter 2-9 which may indicate the value xs (shake-amplitude for table st) if desired.

Similarly, an accelerometer 20-aa is attached to the mass m (that is clamped onto the belt, and its output may be fed to an amplifier 20-1, and then to an amplitude meter 20-3 if desired, to indicate the value xm (shake-amplitude for mass m).

In any event, each accelerometer output is fed, in common, to a phase meter 20-5, which measures the relative phase between the peaks of the (sine wave) motions of the mass and table while shaken. The tuneable sinewave generator 2-1 is adjusted until these relative phases are at 90° degrees.

This 90° degree phase relation is here assumed to occur at "resonance". One example for so measuring phase is with the use of Lisajou figures on an oscilloscope. Another way to measure phase, as well as amplitude, is with an FFT analyzer. These devices are in common use. The frequency at which the 90° degree phase relation occurs is, of course, resonant frequency: the value f_(R) used in the above equation.

Note: the equation above for determining value C has no dependence on belt length. If mass m is clamped at a different position along the belt length, and the measurement process repeated, a different value of c will result. We will show that this should occur by reference (below), to FIG. 3, and will use this as a straight-forward "stress-strain, tension/compression bar model" to derive force-displacement equations in terms of elastic and damping parameters. When compared with the force-displacement equation for the "lumped parameter" model, FIG. 4, it will be seen that any segment of such a belt should have a damping coefficient which is inversely proportional to belt length (i.e.,c˜1/L).

Analysis per Tension-Compression BAR Model (FIG. 3):

FIG. 3 may be understood as a "BAR model" for analyzing forces F,F' stressing a subject length of such a belt B in tension or compression. The segment will be assumed to have a length L with a test mass m disposed therealong at varied positions x, and the resultant displacements (under sinusoidal shaking at resonance) will be U_(o), U_(e) at respective ends.

Assuming that strain ε is defined as du/dx, a displacement U may be defined as the integral; ##EQU2##

Stress α is given as: α=E ε+qdE/dt, or α=F/A

Where

ε=elastic modulus,

q=damping coefficient for bar

t=time

A=cross-sectional area

Thus, one may represent differential motion, U_(L) -U_(O) as: ##EQU3## above analysis summarized in FIG. 3a.

Workers will here understand that a belt is really a tension-compression bar model. The Test Fixture, here mentioned, yields data for a "Lumped Parameter" model (e.g., see below, and FIG. 4). The following analysis of the Lumped Parameter model shows how the measured damping data can be related to the "BAR" model.

Analysis Via "Lumped-Parameter" model (FIG. 4):

FIG. 4 may be understood as a "Lumped-Parameter" model, to be compared with the BAR model above, Here, cc represents a "dashpot damping coefficient, and K: spring stiffness, with U_(o), U_(e) as before.

At dashpot dd, the damping force F_(cc) may be expressed as: ##EQU4##

At "spring" k, related force F_(k) may be expressed as:

    F.sub.k =K(u.sub.e-u.sub.o)

Total force F is F_(cc) +F_(k)

Hence, the differential displacement u_(e) -u_(o) is:

    u.sub.e -u.sub.o =F/k(i-e.sup.KT/c)

Comparing with the BAR model:

k=AE_(/L)

c=Aq_(/L)

Thus, damping coefficient C is inversely proportional to L, the length of belt being stretched (c˜1/L).

Since the belt in FIG. 1 has two segments, L1 and L-L1, which are undergoing tension/compression, the related damping coefficient c measured is actually the sum of the two different damping coefficients: C₂ /L₁ and C₂ /L-L₁, where C₂ =A_(q). Additionally, there may be damping at the interfaces where the belt is tangent to the pulleys. This is particularly true for toothed synchronous belts where a belt tooth may rub on a pulley groove during this test. This damping is a constant, since it is not dependent on the position of the clamped mass. The measured damping coefficient c can be described as follows:

    c=2*c1+c2/L1+c2/(L-L1);

or

    c=2c.sub.1 +c2/L.sub.1 +c2/L-L.sub.1

To obtain the damping coefficients c1 and c2, tests are run for several positions (several L1 values). If only two positions are used, then one can employ a process for solving two linear equations and two unknowns, and so calculate the constants c1 and c2. If more than two positions are used, then one of several well known "least squares, curve fitting" processes can be used to find the "best" values for c1 and c2. An example of such a process is the Marquardt-Levenbert algorithm. of course, using more test positions gives more accurate values for c₁ and c₂.

Results

This invention will be thus understood to measure damping parameters for resilient belts such that a lumped parameter damping coefficient can be easily calculated for any length of belt (used to transmit motion from one shaft to another).

This is done, principally to determine belt damping coefficients, and so enable an accurate prediction of a belt's dynamic motion performance,--for mechanisms utilizing such belts, so that different belts can be compared for their inherent damping capability, and so that a belt can be "sized" to obtain desirable damping properties.

A salient advantage is that belt damping can be determined without actually testing the belt in the mechanism for using it. Also, one can predict damping for belts of different lengths just by testing one length of belt. A simple test fixture can be used for many different kinds and lengths of belt construction. One can also so test other resilient "belt-like" web materials, such as lengths of: photographic film, magnetic tape, ink ribbon, rope, cables, paper strips, fabric strips, etc.

Advantages Over Past Practice

Does not require "trial-and-error" testing in the actual mechanism; and just one belt-length is needed to determine damping for many different lengths;

Does not require a specific sample size, as virtually any length belt may be used.

Does not require that the belt be permanently altered in any way; or "over-stressed" or tested in actual use-environment.

Related products for using such belts or belt-like materials are: microfilm film advance mechanisms, document transports, document positioning systems, paper advance mechanisms in printers, pen plotters, magnetic and optical digital storage devices, magnetic tape recorders, printhead positioning mechanisms in typewriters and computer printers, printer ribbon advance mechanisms, optical mirror positioning mechanisms, robots, automatic assembly mechanisms, automotive alternator drives, camshaft drives, and air conditioning compressor drives, automatic adhesive tape dispensers. Such can also benefit from this invention.

Reprise

In summary, I teach determining the damping coefficient of a resilient web length by:

1-stretching the length between a pair of pulley means, separated by distance L, and mounted on a relatively fixed frame; with a prescribed test mass m clamped thereon at distance L₁ from a pulley;

2-mounting the frame on a shaker table and providing a shaking means (pref. sinusoidal) to shake the frame and test mass while monitoring/adjusting the frequency thereof to arrive at resonance;

3-deriving accelerometer output (amplitude) from this table as it is shaken, while measuring this amplitude A_(T) at resonance;

4-deriving accelerometer output (amplitude A_(m)) from this test mass m, while measuring this amplitude A_(m) at resonance;

5-adjusting the generator to shake the table and the mass at resonant frequency f_(R) ;

e.g., doing so via phase monitor, imposing orthogonal phase-relation!

6-computing damping constant C₁ and C₂ from measurements at two or more mass positions (e.g., L₁);

7-and so determining a damping coefficient C according to the relation;

    C=2C.sub.1 +C.sub.2 /L.sub.1 +C.sub.2 /L-L.sub.1, etc.

Of course, many modifications to the preferred embodiment described are possible without departing from the spirit of the present invention. The invention is not limited to the particular types of sensors or shakers or mountings. Additionally, some features of the present invention can be used to advantage without the corresponding use of other features.

Accordingly, the description of the preferred embodiment should be to be considered as including all possible modifications and variations coming within the scope of the invention as defined by the appended claims. 

What is claimed is:
 1. Apparatus for dynamically calculating the damping coefficient of a web, this apparatus including: a series of pulley units for mounting the specimen web; shaking means for imparting oscillating shaking to said apparatus at resonant frequency; transducer means for indicating displacement during said shaking; and a test mass arranged to be attached to said web at variable distances L, from one of said pulley units, whereby to shake said apparatus, web and test mass at frequencies including resonance frequency, and so determine said damping coefficient for each said distance L.
 2. The apparatus of claim 1, including a frame affixed on shake-surface, whereby controllable shaking means is applied to so shake said frame and shake-surface with said test mass and said web.
 3. The apparatus of claim 2, where said shaking means is arranged to execute sinusoidal vibration.
 4. The apparatus of claim 3, where shake-transducers are coupled to said test mass and said frame to indicate resonance amplitudes.
 5. The apparatus of claim 1, where phase detect equipment is used to determine resonance.
 6. The apparatus of claim 4, where said web comprises a resilient belt for transmitting motion.
 7. The apparatus of claim 6, where a tuneable sine-wave generator unit is used for said shaking.
 8. The apparatus of claim 7, where a damping coefficient C is derived according to the relation: ##EQU5## Where: pi=3.14159m=mass of the object clamped to the web f_(R) : =frequency of sinusoidal motion at resonance xs:=absolute value of motion amplitude for shaker table, and xm=absolute value of amplitude for motion of mass.
 9. The apparatus of claim 3, wherein one or more transducer arrays are affixed to said test mass and said frame, with the outputs thereof being fed in common to phase detect equipment whereby to indicate shake-amplitude at resonance. 